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Journal of Global Optimization

, Volume 71, Issue 2, pp 237–296 | Cite as

Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property

  • M. V. DolgopolikEmail author
Article

Abstract

In this article we present a general theory of augmented Lagrangian functions for cone constrained optimization problems that allows one to study almost all known augmented Lagrangians for these problems within a unified framework. We develop a new general method for proving the existence of global saddle points of augmented Lagrangian functions, called the localization principle. The localization principle unifies, generalizes and sharpens most of the known results on the existence of global saddle points, and, in essence, reduces the problem of the existence of global saddle points to a local analysis of optimality conditions. With the use of the localization principle we obtain first necessary and sufficient conditions for the existence of a global saddle point of an augmented Lagrangian for cone constrained minimax problems via both second and first order optimality conditions. In the second part of the paper, we present a general approach to the construction of globally exact augmented Lagrangian functions. The general approach developed in this paper allowed us not only to sharpen most of the existing results on globally exact augmented Lagrangians, but also to construct first globally exact augmented Lagrangian functions for equality constrained optimization problems, for nonlinear second order cone programs and for nonlinear semidefinite programs. These globally exact augmented Lagrangians can be utilized in order to design new superlinearly (or even quadratically) convergent optimization methods for cone constrained optimization problems.

Keywords

Augmented Lagrangian Cone constrained optimization Localization principle Saddle point Second order cone Semidefinite programming Semi-infinite programming 

Mathematics Subject Classification

65K05 90C30 

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Authors and Affiliations

  1. 1.Saint Petersburg State UniversitySaint PetersburgRussia
  2. 2.Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSaint PetersburgRussia

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